Theorem nfcnv 5868

Description: Bound-variable hypothesis builder for converse relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypothesis

Ref	Expression
nfcnv.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfcnv	⊢ Ⅎ𝑥◡𝐴

Proof of Theorem nfcnv

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cnv 5666	. 2 ⊢ ◡𝐴 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
2		nfcv 2926	. . . 4 ⊢ Ⅎ𝑥𝑧
3		nfcnv.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfcv 2926	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 5158	. . 3 ⊢ Ⅎ𝑥 𝑧𝐴𝑦
6	5	nfopab 5180	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
7	1, 6	nfcxfr 2924	1 ⊢ Ⅎ𝑥◡𝐴

This proof depends on syntax axioms:  Ⅎwnfc 2911   class class class wbr 5108  {copab 5173  ^◡ccnv 5657

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-cnv 5666

This theorem is used by:  nfrn  5946  nfpred  6335  nffun  6590  nff1  6806  funcnvmpt  7026  nfsup  9480  nfinf  9512  gsumcom2  20240  ptbasfi  23955  mbfposr  26028  itg1climres  26090  nfwsuc  36795  aomclem8  44329  rfcnpre1  46310  rfcnpre2  46322  smfpimcc  48091